Mr Trump came under fire for criticising Fox anchor Megyn Kelly during and after a Republican debate. Asked about Kelly in a CNN interview afterwards, Trump said: "You could see there was blood coming out of her eyes. Blood coming out of her wherever."

But Mr Trump has insisted no apology was necessary, and defended his relations with women.

The real estate mogul and television personality, who remains ahead of his 16 rivals in the race for the 2016 Republican presidential nomination, appeared on Sunday news shows in the USA to rebutt the outrage triggered by his off-the-cuff talk.

"I've had such an amazing relationship with women in business. They are amazing executives. They are killers. They are phenomenal," Mr Trump said on ABC's This Week.

Asked if he thought he had gone too far, Mr Trump said, "No, not at all."

Mr Trump told CNN he had "great relationships" with women and had hired thousands, including many top-level employees, adding, "I cherish women."

It has been a recurring pattern in the campaign: Mr Trump makes an offensive comment, incurs a firestorm of reaction and critics sound the death knell for his campaign.

"They were completely inappropriate and offensive comments. Period," Mr Fiorina said on CNN's State of the Union.

"Thin-skinned"

Mr Fiorina criticised Mr Trump as having become angry at Ms Kelly for questioning his comments about women, including calling those he disliked "fat pigs" and "slobs". "You cannot have a president who is thin-skinned," the former Hewlett-Packard Co CEO said on CBS' Face the Nation.

Mr Trump said his comments about blood coming from Kelly were misconstrued as a reference to menstrual blood or hormonal activity and called that interpretation "deviant."

He said he meant to say "nose, ears" and was implying that she was angry. "She asked me a very, very nasty question," he said on This Week.

Trump has repeatedly attacked his fellow candidates, accusing Mr Perry, the former Texas governor, of wearing glasses to look smarter and calling Mr Graham, a US senator from South Carolina, an "idiot".

On Sunday, he attacked Mr Fiorina on several shows as a failed executive who got fired, then lost a bid for the US Senate by a landslide.

He lashed out at Mr Bush as being "negative on women's health" for his recent comments on funding for women's health programmes that drew immediate condemnation from Democrat Hillary Clinton.

Scott Walker, Donald Trump and Jeb Bush (L-R) Photo: AFP

And he took credit for Fox News' record ratings of 24 million viewers for the prime-time debate, saying of Mr Bush: "The man has no energy."

Mr Trump dismissed his critics as beholden to political correctness and pointed to the opinion polls as proof that he was "winning big all over the place".